# On the theory of the Ranque-Hilsch cooling effect

## Summary

An extended Bernoulli equation, adapted to turbulent flow, is applied to a model of the rotational flow in the Ranque-Hilsch vortex tube. It is shown that on the basis of this model a reasonable quantitative explanation of the cooling effect can be given. The experimental results of Hilsch and of Elser and Hoch agree for the greater part with this theory.

As to the construction of the apparatus a theoretical interpretation is given of the influence of the size of the diaphragm.

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## Abbreviations

c p :

specific heat at constant pressure

c v :

specific heat at constant volume

M :

Mach number

P :

ηc p /λ = Prandtl number

p :

pressure

p 0 :

inlet pressure

p 1 :

pressure at the wall

r :

coordinate in radial direction

r 1 :

R′ :

gas constant per unit mass

s :

vector defined by equation (17);s r ,s ϑ ,s z are the components in cylindrical coordinates

T :

absolute temperature (static)

T 0 :

inlet temperature

T 1 :

temperature atr=r 1

T s :

T1+v2/2c p = stagnation temperature

v :

velocity vector;u,v,w are the components in cylindrical coordinates;u 1 andv 1 are the values ofu andv atr =r 1

z :

coordinate in axial direction

α:

η e c p /λ e

β:

dimensionless quantity defined by equation (22)

γ:

c p /c v

η:

coefficient of viscosity

η e :

coefficient of eddy viscosity

ϑ:

angular coordinate

λ:

thermal conductivity

λ e :

eddy thermal conductivity

μ:

mass fraction of cooled gas

ϱ:

density

ϱ 1 :

value of ϱ atr=r 1

σ:

r/r1

ϕ:

v/v1

∇:

nabla operator

## References

1. 1)

Ranque, M. G., J. Phys. Radium4 (1933) 2.

2. 2)

Hilsch, R., Z. Naturforsch.1 (1946) 208.

3. 3)

Elser, K., and M. Hoch, Z. Naturforsch.6a (1951) 25.

4. 4)

Haar, D. ter, and H. Wergeland, Forh. Kong. Norske Vid. Selskab.20 (1947) 55.

5. 5)

Burkhardt, G., Z. Naturforsch.3a (1948) 40.

6. 6)

Prins, J. A., Ned. T. Natuurk.14 (1948) 241.

7. 7)

Ringleb, F., Z. angew. Math. Mech.20 (1940) 185.

8. 8)

Madelung, E., Ann. Phys.43 (1943) 417.

9. 9)

Rubusin, M. W., and H. A. Johnson, Trans. Amer. Soc. mech. Eng.71 (1949) 383.

10. 10)

Reichardt, H., Z. angew. Math. Mech.20 (1940) 297.

11. 11)

Shepherd, C. B., and C. E. Lapple, Ind. engng. Chem.: 51 (1939) 972;32 (1940) 1246.

12. 12)

Eckert, E., and W. Weise, Forsch. Ing. Wes.13 (1942) 246.

13. 13)

Ackeret, J., Ric. sci.20 (1950) 1926.

14. 14)

Webster, D. S., J. A. S. Refr. Eng.58 (1950) 163.

15. 15)

Fulton, C. D., Ibid. 473.

16. 16)

16) Roy Mac Gee Jr., Ibid. 974.

## Author information

Authors

### Corresponding author

Correspondence to J. J. Van Deemter.

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Van Deemter, J.J. On the theory of the Ranque-Hilsch cooling effect. Appl. sci. Res. 3, 174–196 (1952). https://doi.org/10.1007/BF03184927